ECE 4110: Random Signals in Communications and Signal Processing ECE department‚ Cornell University‚ Fall 2013 Homework 2 Due September 20 at 5:00 p.m. 1. (Chebyshev Inequality) Let X1 ‚ . . .‚ X be independent Geometric random variables with parameters p1 ‚ . . . ‚ p respectively (i.e.‚ P (Xi = k) = pi (1 − pi )k−1 ‚ k = 1‚ 2‚ . . .). Let random variable X to be X= i=1 Xi . (a) Find µX = E[X]. (b) Apply the Chebyshev inequality to upper bound P (|X − µX | > a). Evaluate your upper bound
Premium Probability theory Random variable Variance
DISTRIBUTION 3.1 RANDOM VARIABLES AND PROBABILITY DISTRIBUTION Random variables is a quantity resulting from an experiment that‚ by chance‚ can assume different values. Examples of random variables are the number of defective light bulbs produced during the week and the heights of the students is a class. Two types of random variables are discrete random variables and continuous random variable. 3.2 DISCRETE RANDOM VARIABLE A random variable is called a discrete random variable if its set of
Premium Random variable Probability theory
i=1 n i=1 xi = a n x ¯ where x = ¯ 1 n i=1 xi 4. FALSE If X and Y are independent random variables then: E (Y |X) = E (Y ) 1 5. TRUE If {a1 ‚ a2 ‚ . . . ‚ an } are constants and {X1 ‚ X2 ‚ . . . ‚ Xn } are random variables then: n n E i=1 ai X i = i=1 ai E (Xi ) 6. FALSE For a random variable X‚ let µ = E (X). The variance of X can be expressed as: V ar(X) = E X 2 − µ2 7. TRUE For random variables Y and X‚ the variance of Y conditional on X = x is given by: V ar(Y |X = x) = E Y 2 |x
Premium Standard deviation Variance Random variable
ILO6. compute probabilities for a normal distribution and determine normal scores from specific probability requirements. Syllabus 1. Data collection and sampling Data collection Taking a census Sampling Methods of selection a sample Simple random sample Systematic sampling Stratified sampling Summarizing Data Dot Diagrams Stem-and-Leaf Displays Frequency Distributions Graphical Presentations 2. 3. Statistical Descriptions Measures of Location The mean The weighted mean The median
Premium Normal distribution Probability theory Sample size
APPM 3570 — Exam #1 — February 20‚ 2013 On the front of your bluebook‚ write (1) your name‚ (2) 3570/EXAM 1‚ (3) instructor’s name (Bhat or Kleiber)‚ (4) SPRING 2013 and draw a grading table with space for 4 problems. Do only 4 of 5 problems. On the front of your blue book‚ write down which 4 problems you are attempting‚ if you do more than 4 problems‚ only the first 4 problems done will be graded. Correct answers with no supporting work may receive little or no credit. Start each problem on a new
Premium Random variable Probability theory Cumulative distribution function
Probability Distribution • Confidence Intervals Calculations for a set of variables Open the class survey results that were entered into the MINITAB worksheet. We want to calculate the mean for the 10 rolls of the die for each student in the class. Label the column next to die10 in the Worksheet with the word mean. Pull up Calc > Row Statistics and select the radio-button corresponding to Mean. For Input variables: enter all 10 rows of the die data. Go to the Store result in: and select
Premium Statistics Normal distribution Random variable
SYSTEM SIMULATION AND MODELLING 06CS82 UNIT - 1 INTRODUCTION June 2012 1. List any three situations when simulation tool is appropriate and not appropriate tool. 6 M b. Define the following terms used in simulation i)discrete system ii)continuous system iii) stochastic system iv)deterministic system v)entity vi)Attribute 6M c. Draw the flowchart of steps involved in simulation study. 8M June 2010 1a) What is simulation? Explain with flow chart‚ the steps involved in simulation
Premium Random variable Probability theory Poisson distribution
THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2801 Life Contingencies LN1: Chapter 3 (Actuarial Mathematics): Survival distributions Age-at-death random variable T0 – age-at-death (lifetime for newborn) random variable To completely determine the distribution of T0 ‚ we may use (for t ≥ 0)‚ (1) (cumulative) distribution function: F0 (t) = Pr(T0 ≤ t) (2) survival function: s0 (t) = 1 − F0 (t) = Pr(T0 > t) (3) probability density function: f0 (t) = F0 (t) =
Premium Probability theory Random variable Cumulative distribution function
central The Central Limit Theorem A long standing problem of probability theory has been to find necessary and sufficient conditions for approximation of laws of sums of random variables. Then came Chebysheve‚ Liapounov and Markov and they came up with the central limit theorem. The central limit theorem allows you to measure the variability in your sample results by taking only one sample and it gives a pretty nice way to calculate the probabilities for the total ‚ the average and the proportion
Premium Variance Probability theory Normal distribution
1. A gambler in Las Vegas is cutting a deck of cards for $1‚000. What is the probability that the card for the gambler will be the following? a. A face card – there are 12 face cards in a deck of 52 cards. The probability would be 12/52 b. A queen – there are 4 queens in a deck‚ so the probability would be 4/52 c. A Spade - There are 13 cards of each suit so the probability is 13/52 or ¼. d. A jack of spades - There is only 1 jack of spades in a deck‚ so the probability would be 1/52
Premium Playing card Random variable Probability theory